Should off-duty officers have obligations beyond that of private citizens?

They should not have any such obligations because it would be too hard for them to have a personal life if they still had formal, legal obligations while off duty.

If officers were required to essentially be on duty at all times, for example, they would need to carry weapons with them at all times. If you require an officer to respond to any sort of disturbance they see, even when off duty, you would need to require them to carry their weapons all the time so that they could respond effectively and safely.

For this sort of reason, I would argue that off-duty officers should be encouraged to respond when possible, particularly in emergency situations. But there should be no formal requirement that they do so.